CHAPTER TWO
Frequency Distributions
NOTE TO INSTRUCTORS
In this chapter, instructors should emphasize the
importance of visually representing data. The
chapter describes the different ways of organizing
the data in terms of a frequency distribution. The
various
shapes
of
distributions
are
also
presented. Students often forget the importance of
viusally representing the data that they work
with. As a result, it would be useful to show
students how valuable visual representation can be
by demonstrating how frequency distributions can
be used to aid in getting a quick “snapshot” of
data that are collected.
OUTLINE OF RESOURCES
III. Frequency Distributions
 Discussion Question 2-1
 Discussion Question 2-2
 Discussion Question 2-3
 Discussion Question 2-4
(p.
(p.
(p.
(p.
16)
17)
17)
18)
III. Shapes of Distributions
 Discussion Question 2-5 (p. 18)
 Classroom Activity 2-1: Exploring Shapes of
Distribution (p. 18)
III. Next Steps: Stem-and-Leaf Plot
 Online Resources (p. 19)
 Additional Reading (p. 19)
IV.
Handouts
 Handout 2-1: Exploring Shapes of Distribution (p.
20)
CHAPTER GUIDE
I.
Frequency Distributions
1. When we organize our data which is composed of
raw scores, or data that has not been analyzed,
it is useful to look at the distribution of
scores. The distribution allows us to examine
the pattern of our data.
2. We organize our raw scores into a frequency
distribution, which describes the pattern of a
set of numbers by displaying a count or
proportion for each possible value of a
variable.
3. The best and simplest way to arrange our data
is to use a frequency table, which visually
displays the data so that we can see how often
each value occurs.
4. To create a frequency table, we determine the
range of our scores. Then, we create two
columns. In the first column, add the highest
value to the top of the column and the lowest
value to the bottom. In the second column, mark
the number of times each of these values has
occurred in our data set.
5. Sometimes it is better to use a grouped
frequency table that displays the frequencies
for an interval rather than a specific value. A
grouped frequency table is a better choice than
a frequency table when the data are composed of
continuous interval variables, cover a huge
range, or are very large.
> Discussion Question 2-1
What is the difference between a frequency table and a
grouped frequency table? When would you want to use one
type rather than the other?
Your students’ answers should include:
 A frequency table reports every value in a given
data set, whereas a grouped frequency table
reports intervals or ranges of values.
 A frequency table is used to depict data showing
how often certain values occurred and how many
scores were at each value. A grouped frequency
table is used when the values are either:
a. vast in number (such as when reporting
hundreds of values); or
b. continuous-interval
variables,
which
are
reported using several decimal places; or
c. both vast in
places long.
number
and
several
decimal
6. To create a grouped frequency table, find the
highest and lowest scores in the distribution.
If the highest and lowest values are decimals,
round down. Subtract the lowest score from the
highest score and add one. Next, determine the
number of intervals and best interval size.
List the intervals from lowest to highest in a
column. Then, in the other column, count the
number of values in each interval.
> Discussion Question 2-2
What steps are involved in creating a frequency table? A
grouped frequency table?
Your students’ answers should include:
 To create a frequency table:
a. examine the data;
b. create two columns; in the first column
record the values, putting highest at the top
and lowest at the bottom;
c. tally the occurrence of each value; and
d. record the tallies in the second column.
 To create a grouped frequency table:
a. find the highest and lowest scores;
b. use the full range of data, but round scores
down to whole numbers;
c. determine
number
of
intervals
and
best
interval size;
d. determine which number will be the bottom of
the lowest interval; and
e. list the intervals from highest to lowest and
then count the numbers of scores in each.
7. Another way to organize the data is to use a
histogram. Histograms typically depict interval
data with the interval values on the x-axis and
the frequencies on the y-axis. Each bar
represents the frequencies for each value or
interval.
8. To create a histogram, start with a frequency
table. Draw your x- and y-axis and label them
with your variable of interest. Draw a bar for
each value, centering the bar over that value
on the x-axis. The bar should be as high as the
frequency for that value.
9. Histograms can also be created from a grouped
frequency
table.
Instead
of
values,
the
midpoints of the intervals are listed on the xaxis. The remaining steps are the same as those
that you used when constructing a histogram
from a frequency table.
10. Frequency polygons are another way of visually
representing our data using a line graph, where
the x-axis represents the value (or interval
midpoint)
and
the
y-axis
represents
the
frequency. Frequency polygons are similar to
histograms except that dots are used instead of
bars and a line is used to connect the dots.
> Discussion Question 2-3
What is the difference between a histogram and a frequency
polygon?
Your students’ answers should include:
 A histogram looks like a bar graph and often
depicts interval data, with the values of the
variables represented on the x-axis and the
frequencies represented on the y-axis.
 A frequency polygon is a line graph depicting
interval data. It also represents values on the xaxis and frequencies on the y-axis.
> Discussion Question 2-4
What steps are involved in creating a histogram? A
frequency polygon?
Your students’ answers should include:
 To create a histogram:
a. determine the midpoint for each interval, if
needed;
b. draw and label the x-axis and the y-axis of a
graph; and
c. draw a bar for each value.
 To create a frequency polygon:
a. determine the midpoint for each interval, if
needed;
b. draw and label the x-axis and the y-axis;
c. mark a dot above each value and connect the
dots with a line; and
d. add hypothetical values at both ends of the
x-axis and mark dots for the frequency of 0
for each value to create a grounded shape
rather than a floating line.
II.
Shapes of Distributions
1. A normal distribution refers to a bell-shaped,
symmetrical,
and
unimodal
frequency
distribution.
2. We can also use skewness. Skewness describes
how much one of the tails of the distribution
is pulled away from the center.
3. Both descriptive and inferential statistics
require
normal
distributions.
However,
frequently
the
data
are
not
normally
distributed.
4. When the tail of our distribution extends to
the right, we say that our data are positively
skewed. We typically observe positively skewed
data when there is a floor effect—when a
variable is prevented from taking values below
a certain point.
5. Data can also be negatively skewed meaning
that the tail of our distribution extends to
the left. We may observe negatively skewed data
in the case of a ceiling effect—when a variable
is prevented from taking values above a certain
point.
> Discussion Question 2-5
What is skewness? What is the difference between the two
different types of skewness?
Your students’ answers should include:
 Skewness is the amount that a tail of a
distribution is pulled away from the center.
a. Positively skewed data: The tail of the
distribution extends to the right.
b. Negatively skewed data: The tail of the
distribution extends to the left.
Classroom Activity 2-1
Exploring Shapes of Distribution
In this exercise, students will generate examples
of two variables.
 Have the students predict whether the variables
will be positively skewed or negatively skewed.
 Students can then develop questionnaires in
groups to measure these two variables.
 Have them hand out versions of their
questionnaires in class to see if they were
correct in their predictions.
See Handout 2-1 at the end of this chapter.
III. Next Steps: Stem-and-Leaf Plot
1. The stem-and-leaf plot is a graph that displays
all the data points of a single variable both
numerically and visually. It displays the same
information as a histogram—just in a different
way and with more detail.
2. To create a stem-and-leaf plot, first create
the stem by writing down the first digit for
each number of your data from highest to lowest.
The leaves consist of the last digit for each
score and are added in ascending order.
Online Resources
The following Web site provides a wealth
information on statistics:
http://www.math.yorku.ca/SCS/StatResource.html.
of
For information on good and bad visual graphic
presentations, see:
http://www.math.yorku.ca/SCS/Gallery/milestone/.
Additional Reading
Moore, Thomas L., Ed. (2001). Teaching Statistics:
Resources
for
Undergrad-uates.
Mathematical
Association of America.
This book is an instructor’s manual for
teaching undergraduate statistics that advocates a
hands-on approach.
PLEASE NOTE: Due to formatting, the Handouts are only available in Adobe
PDF®.